Rights statement: This is the author’s version of a work that was accepted for publication in Bulletin des Sciences Mathematiques. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Bulletin des Sciences Mathematiques, 140, 7, 2016 DOI: 10.1016/j.bulsci.2016.03.005
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Hill's spectral curves and the invariant measure of the periodic KdV equation
AU - Blower, Gordon
AU - Brett, Caroline
AU - Doust, Ian
N1 - This is the author’s version of a work that was accepted for publication in Bulletin des Sciences Mathematiques. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Bulletin des Sciences Mathematiques, 140, 7, 2016 DOI: 10.1016/j.bulsci.2016.03.005
PY - 2016/9/26
Y1 - 2016/9/26
N2 - This paper analyses the periodic spectrum of Schrodinger's equation $-f''+qf=\lambda f$ when the potential is real, periodic, random and subject to the invariant measure $\nu_N^\beta$ of the periodic KdV equation. This $\nu_N^\beta$ is the modified canonical ensemble, as given by Bourgain ({Comm. Math. Ph899ys.} {166} (1994), 1--26), and $\nu_N^\beta$ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues $(\lambda_n)$. For $\beta, N>0$ small, there exists a set of positive $\nu_N^\beta$ measure such that $(\pm \sqrt{2(\lambda_{2n}+\lambda_{2n-1})})_{n=0}^\infty$ gives a sampling sequence for Paley--Wiener space $PW(\pi )$ and the reproducing kernels give a Riesz basis. Let $(\mu_j)_{j=1}^\infty$ be the tied spectrum; then $(2\sqrt{\mu_j}-j)$ belongs to a Hilbert cube in $\ell^2$ and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence $(\sqrt{\mu_j})_{j=1}^\infty$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics $\sum_j g(\sqrt{\lambda_{2j}})$ with test function $g\in PW(\pi)$ satisfy Gaussian concentration inequalities.\par
AB - This paper analyses the periodic spectrum of Schrodinger's equation $-f''+qf=\lambda f$ when the potential is real, periodic, random and subject to the invariant measure $\nu_N^\beta$ of the periodic KdV equation. This $\nu_N^\beta$ is the modified canonical ensemble, as given by Bourgain ({Comm. Math. Ph899ys.} {166} (1994), 1--26), and $\nu_N^\beta$ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues $(\lambda_n)$. For $\beta, N>0$ small, there exists a set of positive $\nu_N^\beta$ measure such that $(\pm \sqrt{2(\lambda_{2n}+\lambda_{2n-1})})_{n=0}^\infty$ gives a sampling sequence for Paley--Wiener space $PW(\pi )$ and the reproducing kernels give a Riesz basis. Let $(\mu_j)_{j=1}^\infty$ be the tied spectrum; then $(2\sqrt{\mu_j}-j)$ belongs to a Hilbert cube in $\ell^2$ and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence $(\sqrt{\mu_j})_{j=1}^\infty$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics $\sum_j g(\sqrt{\lambda_{2j}})$ with test function $g\in PW(\pi)$ satisfy Gaussian concentration inequalities.\par
KW - Periodic eigenvalues
KW - Spectral theory
KW - Random operators
KW - Linear statistics
U2 - 10.1016/j.bulsci.2016.03.005
DO - 10.1016/j.bulsci.2016.03.005
M3 - Journal article
VL - 140
SP - 864
EP - 899
JO - Bulletin des Sciences Mathématiques
JF - Bulletin des Sciences Mathématiques
SN - 0007-4497
IS - 7
ER -